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T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).
19

%I #11 Mar 12 2023 14:18:18

%S 1,1,1,2,4,2,5,34,34,5,15,500,2052,500,15,52,10900,278982,278982,

%T 10900,52,203,322768,68162042,455546040,68162042,322768,203,877,

%U 12297768,26419793726,1625686993918,1625686993918,26419793726,12297768,877

%N T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

%C Table starts

%C ...1.........1..............2.................5.................15

%C ...1.........4.............34...............500..............10900

%C ...2........34...........2052............278982...........68162042

%C ...5.......500.........278982.........455546040......1625686993918

%C ..15.....10900.......68162042.....1625686993918.103204230192540988

%C ..52....322768....26419793726.10764437129618296

%C .203..12297768.15002771641712

%C .877.580849872

%H Andrew Howroyd, <a href="/A207868/b207868.txt">Table of n, a(n) for n = 1..231</a> (terms 1..49 from R. H. Hardin)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexColoring.html">Vertex Coloring</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Graph_coloring">Graph Coloring</a>.

%e Some solutions for n=5 k=3

%e ..0..1..0....0..1..2....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0

%e ..1..0..1....1..0..3....1..0..1....1..0..1....1..2..0....1..0..1....1..0..1

%e ..0..1..0....0..1..0....0..1..0....2..1..0....0..1..2....0..2..3....0..1..2

%e ..1..0..1....1..0..1....1..0..1....0..2..3....1..0..1....1..0..1....1..0..1

%e ..0..1..0....0..1..0....2..1..0....1..3..0....2..1..0....0..1..0....0..1..0

%Y Columns 1..5 are A000110(n-1), A207864, A207865, A207866, A207867.

%Y Main diagonal is A207863.

%Y Cf. A207997 (3 colorings), A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings).

%Y Cf. A207981, A208001 (knight), A208021 (king), A208054, A208096, A208301.

%K nonn,tabl

%O 1,4

%A _R. H. Hardin_, Feb 21 2012